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User blog:Tetramur/Possible explanations to tetrational arrays on BEAF
Hello to all! I want to introduce my possible explanation of tetrational arrays of BEAF. I'll provide some examples, of course. If you have something to say to me, I'm always glad to hear y'all. Let's start! Part 1 - Small arrays First of all, I want to look at relatively small array, which is array for Bowers' goplexulus - {10,100 ((1)1) 2}. What is it? It is a stage 100, if stage 1 is second gongulus (100-dimensional hypercube with sidelength 100 filled in with 10's from just one 10), stage 2 is bongulus, stage 3 is trongulus etc - or this is 100-dimensional hypercube with sidelength 100 which is filled in with 100-dimensional hypercubes with sidelength 100 ... another 97 repetitions ... which is filled in with 100-dimensional hypercubes with sidelength 100. What is {10,100 ((1)(1)1) 2}? It is giplexulus - and it is array that grows from the array of goplexulus with the same operations as getting the entire goplexulus from one 10 alone - that is, we describe goplexulus as ONE entry and we build the 100-dimensional hypercube with sidelength 100 from the goplexuli as substage 1, 100-dimensional hypercube from this hypercube as entries - as substage 2, ... giplexulus is substage 100. We'll iterate this operation. So, we have: Stage 1 - {10,100 ((1)1) 2} (array has 100^100^100 entries) Stage 2 - {10,100 ((1)(1)1) 2} (array has 100^100^200 entries) Stage 3 - {10,100 ((1)(1)(1)1) 2} (array has 100^100^300 entries) ... Stage 100 - {10,100 ((2)1) 2} - boplexulus (array has 100^100^100^2 entries). Whoops, this is stage 100 - and we are on a plane separator! That is similar to how at usual plane array - {10,100 (2) 2} - we had 100*100 square of 10s. But we iterate further analogously - and at this time we map one 10 to boplexulus and with all arrays we make all operations that we made when we built the array for boplexulus - from one 10 into boplexulus, from boplexulus into biplexulus etc. Stage 1 - {10,100 ((2)1) 2} (array has 100^100^100^2 entries) Stage 2 - {10,100 ((2)(2)1) 2} - biplexulus (array has 100^100^(100^2*2) entries) Stage 3 - {10,100 ((2)(2)(2)1) 2} - baplexulus (array has 100^100^(100^2*3) entries) ... Stage 100 - {10,100 ((3)1) 2} - troplexulus (array has 100^100^100^3 entries). So, we ran into only two groups. Let's see what happens with 99th group... Stage 1 - {10,100 ((99)1) 2} (array has 100^100^100^99 entries) Stage 2 - {10,100 ((99)(99)1) 2} (array has 100^100^(100^99*2) entries) Stage 3 - {10,100 ((99)(99)(99)1) 2} (array has 100^100^(100^99*3) entries) ... Stage 100 - {10,100 ((100)1) 2} = {10,100 ((0,1)1) 2} (array has 100^100^100^100 entries). Uh-oh, THIS is enormously huge goduplexulus! Part 2. From goduplexulus to higher arrays... This is complicated because when we multipicate, for example, 100th degree with itself, it'll produce only 200th degree. But we are not scared with difficulties, right? Suppose we have an array with 100^100^100 entries. When we raise 100^100^100 into 100^100th degree, we only get 100^100^(100*2). (To connect this to arrays, we should remember that goplexulus has 100^100^100 elements in its array and giplexulus has 100^100^(100*2) elements in its array). And we must iterate until multiplier in the third level raises to 100. That is only one group. The exponent in fourth level now has value 2. Second group raises this multiplier to 3, ..., 99-th group raises it to 100. I described all these operations in part 1. Now we have four levels, each is equal to 100 - and we want to expand this to five levels. How will we do this? 0. Starting value is {10,100 ((0,1)1) 2}. It has 100^100^100^100 entries. 1. Raise 100^100^100^100 to 100^100^100th degree. Multipler 2 now is "sitting" in third level. 2. Continue raising until multipler reach 100. Now we have 100^100^100^101. 3. Raise to 100^100^101th degree (all exponent minus first 100, this is important)... 4. Continue... have 100^100^100^102. 5. Raise... continue... 100th iteration of this process will shift 2 into fourth level (100^100^100^(100*2)). Array for {10,100 ((0,2)1) 2} has this many entries. 6. Continue... we have 100th iteration of THIS (points 1-5) and multiplier will increase to 3 - the whole array is {10,100 ((0,3)1) 2}. 7. 99-th iteration of this will make fifth level to appear, but it has value only 2. In BEAF, the whole array is {10,100 ((0,100)1) 2} = {10,100 ((0,0,1)1) 2}. What happened to array? We have added one zero to internal separator, 8. Repete steps 1-7 98 times (each time one zero is added) - and we finally have array for gotriplexulus at the stage 100 - {10,100 (((1)1)1) 2} or {10,100 ((0,0,0,0,...0,1)1) 2} with 100 zeros. We can define the sixth, seventh level of exponentiation... by the same way. The limit of this is epsilon-zero in FGH - and with tetrational arrays we can't define anything higher, we need pentational arrays. Category:Blog posts